text: Irving Langmuir was the first to derive a scientifically based adsorption isotherm in 1918. The model applies to gases adsorbed on solid surfaces. It is a semi-empirical isotherm with a kinetic basis and was derived based on statistical thermodynamics. It is the most common isotherm equation to use due to its simplicity and its ability to fit a variety of adsorption data. It is based on four assumptions:
  # All of the adsorption sites are equivalent, and each site can only accommodate one molecule.
  # The surface is energetically homogeneous, and adsorbed molecules do not interact.
  # There are no phase transitions.
  # At the maximum adsorption, only a monolayer is formed. Adsorption only occurs on localized sites on the surface, not with other adsorbates.
  These four assumptions are seldom all true: there are always imperfections on the surface, adsorbed molecules are not necessarily inert, and the mechanism is clearly not the same for the very first molecules to adsorb to a surface as for the last. The fourth condition is the most troublesome, as frequently more molecules will adsorb to the monolayer; this problem is addressed by the BET isotherm for relatively flat (non-microporous) surfaces. The Langmuir isotherm is nonetheless the first choice for most models of adsorption and has many applications in surface kinetics (usually called Langmuir–Hinshelwood kinetics) and thermodynamics.
  Langmuir suggested that adsorption takes place through this mechanism: , where A is a gas molecule, and S is an adsorption site. The direct and inverse rate constants are k and k. If we define surface coverage, , as the fraction of the adsorption sites occupied, in the equilibrium we have:
  or
  where  is the partial pressure of the gas or the molar concentration of the solution.
  For very low pressures , and for high pressures .
  The value of  is difficult to measure experimentally; usually, the adsorbate is a gas and the quantity adsorbed is given in moles, grams, or gas volumes at standard temperature and pressure (STP) per gram of adsorbent. If we call v the STP volume of adsorbate required to form a monolayer on the adsorbent (per gram of adsorbent), then , and we obtain an expression for a straight line:
  Through its slope and y intercept we can obtain v and K, which are constants for each adsorbent–adsorbate pair at a given temperature. v is related to the number of adsorption sites through the ideal gas law. If we assume that the number of sites is just the whole area of the solid divided into the cross section of the adsorbate molecules, we can easily calculate the surface area of the adsorbent.
  The surface area of an adsorbent depends on its structure: the more pores it has, the greater the area, which has a big influence on reactions on surfaces.
  If more than one gas adsorbs on the surface, we define  as the fraction of empty sites, and we have:
  Also, we can define  as the fraction of the sites occupied by the j-th gas:
  where i is each one of the gases that adsorb.
  Note:
  1) To choose between the Langmuir and Freundlich equations, the enthalpies of adsorption must be investigated. While the Langmuir model assumes that the energy of adsorption remains constant with surface occupancy, the Freundlich equation is derived with the assumption that the heat of adsorption continually decrease as the binding sites are occupied. The choice of the model based on best fitting of the data is a common misconception.
  2) The use of the linearized form of the Langmuir model is no longer common practice. Advances in computational power allowed for nonlinear regression to be performed quickly and with higher confidence since no data transformation is required.